Strength of strongest dominating sets in fuzzy graphs

Document Type : Original paper


1 Shahrood University of Technology

2 Shahed University

3 Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, iran


A set S of vertices in a graph G=(V,E) is a dominating set of G if every vertex of V-S is adjacent to some vertex of S. For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from some vertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongest dominating set as well as strength of strongest $k$-step dominating set in fuzzy graphs. We determine various bounds for these parameters in fuzzy graphs. We also determine the strength of strongest dominating set in some families of fuzzy graphs including complete fuzzy graphs and complete bipartite fuzzy graphs. 


Main Subjects

[1] E.J. Cockayne, R.M. Dawes, and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980), no. 3, 211–219.
[2] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977), no. 3, 247–261.
[3] G. Dror, A. Lev, and Y. Roditty, A note: some results in step domination of trees, Discrete Math. 289 (2004), no. 1-3, 137–144.
[4] M. Farhadi Jalalvand, N. Jafari Rad, and M. Ghorani, Some results on the exact 1-step domination graphs, Math. Montisnigri 44 (2019), 15–27.
[5] A. N. Gani and V.T. Chandrasekaran, Domination in fuzzy graph, Adv. Fuzzy Sets Syst. 1 (2006), 17–26.
[6] A.N. Gani and P. Vadivel, Contribution to the theory of domination, independence and irrenundance in fuzzy graph, Bulletin of Pure and Applied sciences 28E (2009), no. 2, 179–187.
[7] H. Gavlas and K. Schultz, Efficient open domination, Electronic Notes Discrete Math. 11 (2002), 681–691.
[8] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998.
[9] M.A. Henning and A. Yeo, Total Domination in Graphs, Springer Monographs in Mathematics, 2013.
[10] P. Hersh, On exact n-step domination, Discrete Math. 205 (1999), no. 1-3, 235–239.
[11] O.T. Manjusha and M.S. Sunitha, Strong domination in fuzzy graphs, Fuzzy Inf. Eng. 7 (2015), no. 3, 369–377.
[12] D.A. Mojdeh and B. Ashrafi, On domination in fuzzy graphs, Advances in Fuzzy Math. 3 (2008), no. 1, 1–10.
[13] D.A. Mojdeh and M.M. Pourpasha, Fuzzy defining number of the edge fuzzy graphs, Int. Math. Forum 2 (2007), no. 17-20, 893–904.
[14] J.N. Mordeson and P.S. Nair, Fuzzy Graphs and Fuzzy Hypergraphs, vol. 46, Physica-verlage, Heidelberg, New York,, 2012.
[15] A. Rosenfeld, Fuzzy Graphs, in: Fuzzy Sets and Their Application to Cognitive and Desision Processes, 1975, pp. 77–95.
[16] K.L. Schultz, Step Domination in Graphs, Ph.D. thesis, Western Michigan University, 1995.